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$\ldots$

$$\begin{aligned} \exp\big( -t^a \wedge \iota_a \big) \big( d_{dR} + d_W \big) \exp\big( t^a \wedge \iota_a \big) & = d_{dR} + d_W + \big[ d_{dR} + d_W, t^a \wedge \iota_a \big] + \tfrac{1}{2} \Big[ \big[ d_{dR} + d_W, t^a \wedge \iota_a \big], t^a \wedge \iota_a \Big] \\ & = d_{dR} + d_W \\ & \phantom{=} + \underset{ - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c + r^a }{ \underbrace{ \big[ d_W , t^a \big] } } \wedge \iota_a - t^a \wedge \underset{ = \mathcal{L}_{a} }{ \underbrace{ \big[ d_{dR} + d_W, \iota_a \big] } } \\ & \phantom{=} + \tfrac{1}{2} \Big[ \underset{ \mathclap{ - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c \iota_a + r^a \iota_a - t^a \mathcal{L}_a } }{ \underbrace{ \big[ d_{dR} + d_W, t^a \wedge \iota_a \big] } } , t^a \wedge \iota_a \Big] \end{aligned}$$

(wait…)

a circle-principal bundle?

An Ehresmann connection

$$\theta^5 \;\in\; \Omega^1\big( \Sigma^6 \big)$$

$$H \;\in\; \Omega^3\big(\Sigma^6\big)$$

$$H = d B$$

$$H \;=\; \star_6 H$$

$$\tilde H \coloneqq \star_5 H \coloneqq \star_5 \big( H - \mathcal{F} \wedge d x^5 \big)$$

$$A \;\coloneqq\; \iota_5 B$$

$$B \;=\; A \wedge d x^5 + B^{\mathrm{bas}}$$

$$F \coloneqq d_5 A$$

$$\begin{aligned} \mathcal{F} &\coloneqq \iota_5 H \\ & = \iota_5 d B \\ & = -d \iota_5 B + [\iota_5, d] B \\ & = - d A + \mathcal{L}_5 B \\ & = - F - \mathcal{L}_5 A \wedge d x^5 + \mathcal{L}_5 ( A \wedge d x^5 + B^{basic} ) \\ & = - F + \mathcal{L}_5 B^{bas} \end{aligned}$$

$$\star_6 H = \star_6 \big( \mathcal{F} \wedge d x^5 + (H - \mathcal{F} \wedge d x^5) \big) = \star_5 \mathcal{F} + \tilde H \wedge d x^5$$